If You Know the Average You Can Find a Missing Number

Yes. The mean tells you the total, and subtracting the values you do have leaves the one you do not. If six observations have a mean of 15, their total must be $6\times 15=90$ — and the gap is the missing value.

Running the mean backwards like that is the most useful trick in the chapter. This page covers everything in the ICSE Class 7 Mathematics chapter's second part: the mean and missing observations, the median, the mode, and bar graphs.

The median is the middle value once the data is arranged in ascending order — and arranging first is not optional.

Odd number of observations. The middle one is the median. For $12,15,18,20,25$ there are 5 values, so the median is the 3rd:

$$\text{Median}=18$$

The position is given by $\frac{n+1}{2}$, so for $n=5$ it is the 3rd value, and for $n=9$ the 5th.

Even number of observations. There is no single middle, so take the mean of the two middle values. For $12,15,18,20$ there are 4 values, so the median is the mean of the 2nd and 3rd:

$$\text{Median}=\frac{15+18}{2}=16.5$$

Another. For $8,11,14,16,19,23$ with $n=6$, the median is $\frac{14+16}{2}=15$.

The median splits the data in half — as many observations below it as above.

The commonest error is reading the middle of the unsorted list. For the raw data $23,45,12,45,30$ the middle entry as written is 12, but sorted it becomes $12,23,30,45,45$ and the median is 30 — a completely different answer from the same five numbers.

The mode is the observation that occurs most often.

Worked example. For $2,3,3,4,5,3$ the value 3 appears three times, more than any other, so

$$\text{Mode}=3$$

For the shoe sizes $5,6,7,5,6,6,7,8,5,6$, the frequencies are 5 three times, 6 four times, 7 twice and 8 once — so the mode is 6.

Data can have more than one mode. For $2,2,3,4,4,5$ both 2 and 4 occur twice, which is more often than anything else, so there are two modes, 2 and 4. Such data is called bimodal.

And data can have no mode at all. In $1,2,3,4,5$ every value occurs once, so no observation is more frequent than the others.

The mode is the only one of the three averages that works for non-numerical data — a shopkeeper cannot average shirt colours, but can certainly find the colour sold most often. That is exactly why shops stock by mode: the most popular size, not the average size.

A frequency table makes the mode obvious, since it is simply the value with the highest frequency — which is why building the table first turns this into a one-glance answer.

A bar graph shows data as bars of equal width with equal gaps, where the height represents the value against a stated scale.

Choosing the scale is the step that needs care. Divide the largest value by the height available. If the largest value is 40 and you have 8 cm of graph paper:

$$\text{scale: }1\text{ cm}=\frac{40}{8}=5\text{ units}$$

so the tallest bar is 8 cm and a value of 25 is drawn as $\frac{25}{5}=5$ cm.

Drawing it needs four things, each marked separately:

- Both axes labelled with the quantity they show
- The scale stated in words, as 1 cm = 5 students
- Bars of equal width with equal gaps
- A title naming what the graph shows

Reading it. To find a value, read the bar's height against the scale. A bar 6 cm tall on a scale of 1 cm = 5 units represents $6\times 5=30$.

A school notice board comparing house points is a bar graph in daily use.

Choose the scale before drawing a single bar. A scale picked afterwards usually leaves the tallest bar running off the page or squashed into 2 cm, and either way the graph has to be redrawn — whereas one division at the start settles it.